by Josef A

Created on: June 14, 2009

A more advanced econometric approach is the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model. It is an extension of the Autoregressive Conditional Heteroskedasticity (ARCH) model which is often not sufficient to capture the dynamics of the conditional volatility. A variety of extensions and modifications exist for GARCH approaches depending on the specific empirical question that is analyzed. GARCH models are mostly applied and developed in the field of finance, but can help to get closer insight whenever volatility is not constant over time (Heteroskedasticity). The concept and the principal ideas are described in the following.

The conditional volatility can either be modeled as the conditional variance or the conditional standard deviation. For simplicity the case of conditional variance shall be assumed. The model for the mean can be any kind of simple or complex specification. The obtained error terms (residuals) are used to explain the conditional variance by taking the squared lagged values of the error term series. In addition, a constant is usually included in the model. So far this approach is the same as an ARCH process. It is sometimes very unpractical to use ARCH models as the number of required lagged residuals can be very high. Hence, an additional explanatory variable is included: past conditional variances. Usually, one lag of the conditional variance is enough to capture the dynamics. In general, a GARCH (1,1), that means one lagged squared residual and one lagged conditional variance is included, is found to be sufficient. It is not possible to use Ordinary Least Squares (OLS) when estimating GARCH models, therefore methods such as Maximum Likelihood (ML), Quasi Maximum Likelihood (QML), and Generalized Method of Moments (GMM) are used to obtain estimates.

In principle, any additional variable can be added to a standard GARCH model in order to explain the conditional volatility. Well-known modifications of the standard GARCH model are GARCH-in-mean (GARCH-M) including the conditional variance in the mean equation and the exponential GARCH (EGARCH) model trying to measure asymmetric effects. When modeling the variance, researches have to cope with similar problems as in regular time series, namely with unit roots. To account for unit roots, approaches such as integrated GARCH models (IGARCH) or fractionally integrated GARCH (FIGARCH) models have been put forward. Many more modifications have emerged attempting to cope with problems of specific empirical analyses. GARCH models are generally quite flexible and have become a standard tool in most software packages.

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